Numerical implementation of the method of fictitious domains for elliptic equations

Numerical implementation of the method of fictitious domains for elliptic equations

Almas N. Temirbekov

Citation: AIP Conference Proceedings 1759, 020053 (2016); doi: 10.1063/1.4959667 View online: http://dx.doi.org/10.1063/1.4959667

View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1759?ver=pdfcov Published by the AIP Publishing

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Numerical implementation of the method of fictitious domains for elliptic equations

Almas N. Temirbekov

Al-Farabi Kazakh National University, Almaty, Kazakhstan

Abstract. In the paper, we study the elliptical type equation with strongly changing coefficients. We are interested in studying such equation because the given type equations are yielded when we use the fictitious domain method. In this paper we suggest a special method for numerical solution of the elliptic equation with strongly changing coefficients. We have proved the theorem for the assessment of developed iteration process convergence rate. We have developed computational algorithm and numerical calculations have been done to illustrate the effectiveness of the suggested method.

Keywords: Elliptical equation, Dirichlet problem, The equation with rapidly changing coefficients, Computational algorithm, Iterative process, Fictitious domain method, Boundary conditions

PACS:02.30.Jr

INTRODUCTION

It is efficient to use the fictitious domain method in irregular shape domains for numerical solutions of elliptical type elliptical type equations.

Reference [1] suggests efficient (due to operations number) difference scheme of the second order accuracy, alternately-triangular scheme for numerical solution of the elliptical equation. Modified alternately-triangular iteration method with Chebyshev parameters of Dirichlet differential problem solution for the elliptical equation of the second- order accuracy was made in reference [2]. V. I. Lebedev in his monograph [3] studied the use of composition method for the solution of problems on characteristic constants, nonstationary problems, Dirichlet problems for biharmonic equation and domain problems. Reference [4] studies stationary differential problem for Poisson’s equation with piecewise constant coefficients in subdomains. Poisson’s equation on the boundary approximates in a specific way, that is, the differential equation coefficients are selected as quotient which denominator contains coefficients sum in subdomains. Two-phase iteration process based on domain partitioning method has been built.

Solving the Poisson equation for pressure is the main computing unit in the problems of hydrodynamics of an incompressible fluid. In [5], a parallel implementation of the method of fictitious domains for the Poisson’s equation in a three-dimensional region with a stepped-back is proposed. This method is based on the parallel implementation of the fast algorithm for solving the Poisson equation in a parallelepiped. In [6], a variant of the method of collocation and least residuals for the numerical solution of the Poisson equation in polar coordinates on a non-uniform grid is proposed. In [7, 8], the method of fictitious domains is used for the numerical solution of elliptic equations with complex geometry. Method of fictitious domain for the equations of mathematical physics was studied by A. N.

Bugrov, A. N. Konovalov, S. S. Smagulov, M. K. Orunhanov, R. Glowinski, V. Girauit and others [5–10]. In these studies various modifications of the method of fictitious domain for the Navier-Stokes equations were investigated.

In the present paper, we suggest a specific method for numerical solution of the elliptical equation with strongly changing coefficients. The basis of the suggested method is in the special replacement of variables which brings the problem with second order discontinuous coefficients to the problem with first order discontinuous coefficients.

We have built the iteration process with two parameters which takes into account the equation coefficient ration in subdomains. We have proved the theorem for the developed iteration process convergence rate assessment. We have developed computational algorithm and made numerical calculations to illustrate the efficiency of the suggested method.

PROBLEM SETTING

Let Ω be a bounded domain in R² with a piecewise-smooth boundary ∂Ω. For determination, let Ω = Q₁^∪Q₂, Q₁^∩Q₂=Γ,Q₂be strictly internal subdomain. We will study the elliptical equation

−div(k∇u) =f(⃗x), ⃗x∈Ω, (1)

inΩwith boundary condition

u(⃗x) =0, ⃗x∈∂Ω, (2)

where

k(⃗x) =

{ k₁=const, ⃗x∈Q₁, k₂=const,(⃗x), x∈Q₂.

Function f(⃗x)is suggested as belonging to Hilbert space of real functionsL₂(Ω)and are determined in subdomains with the following ways

f(⃗x) =

{ f⁽¹⁾(x), x∈Q₂, 0, x∈Q1.

We will make the replacement of variablesu=2ν^/k1in (1), simple transformations, and get

∆ν^+div(ω∇ν^{) =}−f(⃗x), (3) whereω^=2k(x)_k1 −1. Let’s designateθ^=2k_k1²−1.

We will introduce symbol⃗p=

(ω_∂^∂ν_x1^, ω_∂^∂ν_x2⁾and equation (3) will be written as equation system







∆ν⁺∇⃗p=−f(⃗x), p₁/ω−_∂^∂ν_x

1 =0, p₂/ω−_∂^∂ν_x

2 =0.

(4)

COMPUTATIONAL ALGORITHM

For the numerical solution of equation system (4) with boundary conditionsν|_∂Ω=0,let’s study the iteration method Bνtⁿ⁺¹+∆hνⁿ⁺¹⁺∇h⃗pⁿ⁺¹=−f(⃗x), β^(⃗pn+1−⃗pⁿ) +⃗pⁿ⁺¹

ω ^−∇hν^{n+1=0, (5)} whereBis iteration method operator,β is iteration parameter, indexhmeans difference analogue of differentiation operator. OperatorBin iteration method (5) is selected by the following way

B= (1−τ⁾∆h−τ^div(ρ∇h), (6)

whereρ^{= (}β^+1/ω^)−1. Let’s suppose thatν⁰∈W^o

1

2(Ω)andp⁰=∇q, whereq∈W^o

1

2. This condition is satisfied if(ν^{0,p0) =0.}

Further, we will consider thatB>0 onW^o

1

2. It is enough to have the inequality 1−τ−τ

β ^{>0. (7)}

In case ofBonW^o

1

2, it satisfies operator inequality

−χ1∆≤B≤ −χ2∆. (8)

Constantsχ1andχ2may be selected independent uponθ≥1. Substituting operatorBdetermined as (6) and (5), we get

∆hν^{n+1=F(x), (9)}

⃗pⁿ⁺¹=βρ^⃗pn+ρ∇hν^{n+1, (10)}

where

F(⃗x) = (1−τ⁾∆hνⁿ−τ^div(βρ∇hνⁿ⁾−τ^divh(βρ^{⃗pn). (11)} We present numerical algorithm of method (9), (10). One step of iteration method (9), (10) consists in finding value νⁿ⁺¹due to knownν^{n, ⃗pn}. For this, it is necessary to solve Dirichlet problem for Poisson’s equation (9) inΩ. After that value⃗pⁿ⁺¹upon the known⃗pⁿandνⁿ⁺¹is counted using formula (10).

CONVERGENCE STUDYING

We assess convergence rate method (9), (10). Let’s designate {y,r} ={yⁿ,rⁿ} ={ν−ν^n,p−pⁿ}, {y,ˆ rˆ} = {yⁿ⁺¹,rⁿ⁺¹}.Then equations (5) can be rewritten as

(By_t,ν^{) + (}∇hy,ˆ∇hν^{) + (}∇hr,ˆν^{) =0}∀ν∈W^o

1

2, (12)

βτ^rt+r/ˆ ω−∇hyˆ=0, (13) where{y⁰,r⁰} ∈W^o

1

2×L₂. Herey_t= (yˆ−y)/τ^.

Let’s call functionψ^{from asL}2piecewise gradient if it can be presented as

ψ⁼∇g_i in Q_i; where g_i∈W₂¹(Q_i), (14) g_i|_∂Ω^∩_∂Ωi=0, i=1,2, ...,N

and call functionψgradient if it is as the following

ψ⁼∇g in Ω, where g∈W^o

1 2(Ω).

Asp⁰=∇g, g∈W^o

1

2andω-is piecewise constant,r⁰is piecewise gradient.

Let’s multiply both parts of equation (13) scalarly inL₂on 2τ^{rˆand put}ν⁼²τ^yˆin correlation (12). Adding up the obtained equalities, we have

∥yˆ∥²B− ∥y∥²B+τ²∥y_t∥²B+2τ∥∇hyˆ∥²+βτ∥rˆ∥²−βτ∥r∥²+βτ³∥r_t∥²+2τ

ω^{∥rˆ∥2=0. (15)} Now, we will studyrⁿ. Since

τˆ⁼ β β^+1/ω^r+

1 β^+1/ω^∇y,ˆ

andris piecewise gradient function, ˆτis also piecewise gradient. Thus, allrⁿare piecewise gradient.

LetGbe the space of piecewise gradient functions,G1be the space of gradient functions. It is obvious thatG1⊆G.

We will show that there is rigid embeddingG₁⊂G and find the orthogonality in L₂ of add-insG₁ toG. If ψ ^is orthogonal inL₂to all elementsG₁, we have(ψ^,∇q)_Ω=0 for any element∇q∈G₁. If function∇gis smooth enough and has carrier inQi, then

(ψ^,∇g)_Ω= (ψ^,∇g)_Qi =−(divψ^,g)Qi =−(∆g_i,g)_Qi=0.

Due tog-is arbitrary, the last correlation means that

∆g_i=0in Q_i. (16)

It is evident that the correlation is done in eachQ_i, i=1,2. Thus, elementψ∈G, orthogonal to all elementsG₁, will be presented as (14), whereq_i is harmonic inQ_ifunction. We will find the conditions to be satisfied by functionψ^, orthogonal toG₁onΓ.

Let’s∇q∈G₁, then

0= (ψ^,∇q)_Ω= (∇q₁,∇q)_Q1+ (∇q₂,∇q)_Q2=

∫ Γ

g∂^q1

∂ⁿ1

ds+

∫ Γ

g∂^q2

∂ⁿ2

ds=

∫ Γ

g (∂^q1

∂ⁿ1−∂^q2

∂ⁿ1

) ds

wheren_i are inner normal vectors on∂^Qi. Therefore, values of normal compoundsψ1=∇q₁andψ2=∇q₂ onΓ coincide. Thus, normal compound of vector-functionψ is continuous (in integral sense) at the transfer through Γ. It follows that orthogonal inL₂ add-insG₂ of spaceG₁toG consists of all vector-functions of type (14), normal compound of which is continuous during the transfer through adjacent boundaries, and the forming functionsg_iare harmonic inQ_i.

We continue to study iteration method convergence (9), (10). It was stated that ˆr∈G. Let’s present ˆras ˆr=qˆ+h,ˆ where ˆq∈G₁and ˆh∈G₂. Then, correlation (12) is as the following

(By_t,ν^{) + (}∇y,ˆ∇ν^{) + (q,ˆ} ∇ν^{) + (h,ˆ} ∇ν^{) =0 (17)} for allν∈W^o

1 2.

The last scalar product in (17) equals nought because∇ν∈G₁. Having divided both parts (17) in∥∇ν∥and having assessed the term containing ˆq, we get

|(q,ˆ ∇ν⁾|

∥∇ν∥ ≤|(By_t,ν⁾|

∥∇ν∥ +|(∇y,ˆ∇ν⁾|

∥∇ν∥ ≤√

χ2∥y_t∥B+∥∇hyˆ∥.

As soon as the right part of this inequality is independent upon ν ∈W^o

1

2, and ˆq∈G₁, that is, it is presented as ˆ

q=∇g(g∈W^o

1

2), taking sup onνin the left part of the inequality, we get the assessment

∥qˆ∥ ≤√

χ2∥y_t∥B+∥∇hyˆ∥,

where∥∇hyˆ∥=∥yˆ∥1. Let’s square both parts of this inequality and assess the right part

∥qˆ∥²≤2(χ2∥y_t∥²B+∥∇hyˆ∥²).

Multiplying the last equation byβτ²µ⁽λ^>0 is arbitrary) adding to (15), we have

∥yˆ∥²_B+τ²⁽¹−2βλ χ2)∥y_t∥²_B+2τ⁽¹−βτλ⁾∥∇hyˆ∥² +βτ²λ∥qˆ∥²+2τ^(r,ˆ _ω^rˆ)+βτ∥rˆ∥²≤ ∥y∥²_B+βτ∥r∥².

(18)

Let’s assess scalar product(r,ˆr/ˆ ω^{). For any}δ^{, 0<}δ ^<1 the following inequalities are true (rˆ

ω^,rˆ )

≥ (qˆ

ω^,qˆ )

+ (hˆ

ω^,hˆ )

−2 (

ˆ q ω^,hˆ

)

≥ (1−δ⁾ (hˆ

ω^,hˆ )

+ (

1−1 δ

)(qˆ ω^,qˆ

)

≥ (1−δ⁾ [

∥hˆ∥²_Q1+1 θ^∥hˆ∥2Q2

] +

( 1−1

δ )

∥qˆ∥². (19) Due to ˆh∈G, the following estimate is valid:

∥hˆ∥Q2≤c₃∥hˆ∥²Q₁.

Thus,

∥hˆ∥²_Ω=∥hˆ∥²Q₁+∥hˆ∥²Q₂≤(1+c₃)∥hˆ∥²Q₁. Therefore, we get the following inequality from (19):

(rˆ ω^,rˆ

)

≥c₄(1−δ⁾∥hˆ∥²+ (

1−1 δ

)

∥qˆ∥², c₄= (1+c₃)⁻¹. Using the last inequality, we will take (18) and obtain

∥yˆ∥²_B+τ²⁽¹−2βλ χ2)∥y_t∥²_B+2τ⁽¹−βτλ⁾∥yˆ∥²₁ +βτ∥rˆ∥²+βτ²λ∥qˆ∥²+2τ⁽¹−δ^)c4∥hˆ∥²

+2τ⁽¹−¹_δ)

∥qˆ∥²≤ ∥y∥²_B+βτ∥r∥².

(20)

Let’s fixβ^>0 and selectτ^>0 so that for allθ^>1 the conditionβ ^>0 were satisfied. Let’s selectλ satisfying the conditions

1−2βλ χ2>0, 1−βτλ^>0 and supposeδ⁼_4+βτλ^{4 <1.}

Then,

βτ²λ⁺²τ⁽¹−1/δ^{) =}βτ²λ−2τβτλ

4 =βτ²λ

2 , 1−δ ⁼ βτλ 4+βτλ^. Inequality (20) at suchδ is the following

( 1+c₅τ

χ2

)

∥yˆ∥²_B+βτ^(1+c6τ⁾∥rˆ∥²≤ ∥y|²_B+βτ∥r∥², (21) wherec₅=τ⁽¹−2βλ χ2), c₆=min

{λ 2,₄₊^2c_βτλ^4λ

}

it is easy to see that constantsβ^,τ^,χ2,λ can be selected the same for allθ^{, 1}≤θ≤∞. Thus, the following theorem is proved.

Theorem 1 For anyβ^>0, there isτ⁼τ⁽β⁾independent uponω≥1such that−χ1∆≤B≤ −χ2∆atτ≤τ^constants χ1,χ2are independent uponω^.

In this case iteration process (9), (10) converges with geometric sequence rate, and convergence rate is independent uponω^.

NUMERICAL CALCULATIONS

Test problem (1)-(2) is solved with the above described method. Subdomain Q₂ was selected as a square Q₂= {x_1,k1≤x₁≤x_1,k2;x_2,m1≤x₂≤x_2,m2},wherex_1,k1=0.25, x_1,k2=0.75, x_2,m1=0.25, x_2,m2=0.75. DomainΩcover subdomainQ₂,Ω={0≤x₁≤1; 0≤x₂≤1}.SubdomainQ₁is determined asQ₁=Ω\Q₂the right part is given inQ₂with the following way f(x₁,x₂) =2(x²₂−(x_2,m1+x2,m₂)x₂+x2,m₁x2,m₂) +2(x²₁−(x_1,k1+x_1,k2)x₁+x_1,k1x_1,k2), wherex_1,k1=0.25, x_1,k2=0.75, x_2,m1 =0.25, x_2,m2 =0.75.

Functionf(x₁,x₂) =0 in subdomainQ₁. Iteration parameterτwas selectedτ⁼¹⁰⁻³÷10⁻⁵, and parameterβ ^was determined so that it satisfies condition (7). And it is necessary to watch the parameter symbolω in the subdomains because−1≤ω≤1.

The set problem of elliptical type with strongly changing coefficients was solved by fictitious domain method continued with leading coefficients. Figures 1-2 show the corresponding results of exact and approximate solution having mesh points 101x101.

We used uniform mesh 101x101, 501x501, 1001x1001 for our calculations. The computing experiment on the small mesh numerical experiment was done on the super computer URSA on the basis of 128 four-core processors Intel Xeon series E5335 2.00GHz at Al Farabi KazNU. The developed method is based on building computational algorithm for the elliptical equation with strongly changing coefficients. The developed algorithm uniformly converges at a certain iteration quantity, and the results are accurate within 10⁻¹⁰. Numerical computation results are presented by graphics editor Surfer.

FIGURE 1. The diagram of exact solution having mesh points 101x101.

FIGURE 2. The diagram of approximate solution having mesh points 101x101.

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